Problem: Simplify the following expression: $z = \dfrac{-48r^3 + 12r^2}{24r^3 - 66r^2}$ You can assume $r \neq 0$.
Answer: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-48r^3 + 12r^2 = - (2\cdot2\cdot2\cdot2\cdot3 \cdot r \cdot r \cdot r) + (2\cdot2\cdot3 \cdot r \cdot r)$ The denominator can be factored: $24r^3 - 66r^2 = (2\cdot2\cdot2\cdot3 \cdot r \cdot r \cdot r) - (2\cdot3\cdot11 \cdot r \cdot r)$ The greatest common factor of all the terms is $6r^2$ Factoring out $6r^2$ gives us: $z = \dfrac{(6r^2)(-8r + 2)}{(6r^2)(4r - 11)}$ Dividing both the numerator and denominator by $6r^2$ gives: $z = \dfrac{-8r + 2}{4r - 11}$